复微分几何与其应用  

Complex differential geometry and its applications

作　　者：莫毅明[1] Ngaiming Mok(Department of Mathematics,the University of Hong Kong,Hong Kong,China)

机构地区：[1]香港大学数学系,中国香港

出　　处：《科学通报》2022年第32期3737-3752,共16页Chinese Science Bulletin

基　　金：香港研究资助局优配研究金(17301518,17304321)资助。

摘　　要：《复微分几何与其应用》源自本人早期对有界对称域?的有限体积商空间XΓ:=?/Γ以及对偶Hermite紧型对称空间S的研究.本人解决广义Frankel猜想的论文揭示了极小有理切线簇(variety of minimal rational tangents, VMRT)对单直纹射影流形(X, K)的几何意义,与Hwang合作建立了一套通过VMRT结构π:C (X)→X与其万有族ρ:U→K发展出来的微分几何理论,用以解决包括有理齐性空间G/P的K?hler形变刚性与Lazarsfeld问题等的经典难题,并建立了关于保持VMRT局部双全纯映照的Cartan-Fubini延拓原则,后来Hong和Mok(2010)以及Mok和Zhang(2019)又发展了非同维Cartan-Fubini延拓原则以及子VMRT结构的延拓理论,并且证明了Schubert与Schur刚性定理. VMRT理论同时提示了如何研究?的代数子簇Z??到XΓ的投影.运用Mok和Zhong关于有限体积完备K?hler流形的紧致化定理,本人证明了对秩等于1的任意格成立的AxLindemann定理.对于Shimura簇,即当Γ为算术格时, o-极小结构理论与Hodge理论提供了研究XΓ的非常有效的工具.在此等理论的技巧与研究成果的基础上,本人从复微分几何以及代数几何的视角与Pila及Tsimerman合作,成功证明了期待已久的Shimura簇上的Ax-Schanuel定理.后者与其多方面的推广,为数论里一系列猜想提供了强而有力的研究手段.“Complex differential geometry and its applications” originated from the author’s prior work in K?hler geometry revolving around bounded symmetric domains ?, their finite-volume quotients XΓ:= ?/Γ and dual Hermitian symmetric spaces S of the compact type. The author’s solution of the generalized Frankel conjecture has revealed the importance of the collection of tangents to minimal rational curves in the curvature characterization of S. Together with Hwang, the author has developed the foundation of a differential-geometric theory of uniruled projective manifolds(X, K) based on the variety of minimal rational tangents(VMRT), encapsulated in the VMRT structure π : C(X) → X and the universal family ρ : U → K,enabling them to resolve classical problems in algebraic geometry such as the deformation rigidity of rational homogeneous spaces G/P of Picard number 1, the Lazarsfeld problem and the characterization of uniruled projective manifolds equipped with reductive G-structures. Hwang-Mok established the Cartan-Fubini extension principle for uniruled projective manifolds(X, K) of Picard number 1 under very mild conditions for the analytic continuation of VMRT-preserving local biholomorphisms, which was further developed by Hong-Mok resp. Mok-Zhang to a non-equidimensional version of Cartan-Fubini extension principle resp. analytic continuation of germs of complex submanifolds inheriting sufciently non-degenerate sub-VMRT structures, resulting in particular in Schubert and Schur rigidity theorems. The geometric theory of VMRT has also motivated the author to examine algebraic subsets(defined via the Borel embedding) Z ? ?, and their images πΓ(Z) under the quotient map πΓ: ? → XΓ, i.e., to examine an irreducible component K of the Chow scheme Chow(S), its universal family ρ : U → K and evaluation map μ : U → X, together with the quotient UΓ:= U0/Γ of the restriction U0:= U|?. The fibered space UΓmay be regarded as the support of a meromorphic distribution F, such that the image under the

关 键 词：单直纹射影流形 极小有理切线簇(VMRT) Cartan-Fubini原则 有界对称域 Shimura簇 非寻常交集 

分 类 号：O186.1[理学—数学]